Question: Solve for $x$ : $ 8|x - 10| - 9 = -4|x - 10| + 6 $
Explanation: Add $ {4|x - 10|} $ to both sides: $ \begin{eqnarray} 8|x - 10| - 9 &=& -4|x - 10| + 6 \\ \\ { + 4|x - 10|} && { + 4|x - 10|} \\ \\ 12|x - 10| - 9 &=& 6 \end{eqnarray} $ Add ${9}$ to both sides: $ \begin{eqnarray} 12|x - 10| - 9 &=& 6 \\ \\ { + 9} &=& { + 9} \\ \\ 12|x - 10| &=& 15 \end{eqnarray} $ Divide both sides by ${12}$ $ \dfrac{12|x - 10|} {{12}} = \dfrac{15} {{12}} $ Simplify: $ |x - 10| = \dfrac{5}{4}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 10 = -\dfrac{5}{4} $ or $ x - 10 = \dfrac{5}{4} $ Solve for the solution where $x - 10$ is negative: $ x - 10 = -\dfrac{5}{4} $ Add ${10}$ to both sides: $ \begin{eqnarray} x - 10 &=& -\dfrac{5}{4} \\ \\ {+ 10} && {+ 10} \\ \\ x &=& -\dfrac{5}{4} + 10 \end{eqnarray} $ Change the ${ + 10}$ to an equivalent fraction with a denominator of $4$ $ x = - \dfrac{5}{4} {+ \dfrac{40}{4}} $ $ x = \dfrac{35}{4} $ Then calculate the solution where $x - 10$ is positive: $ x - 10 = \dfrac{5}{4} $ Add ${10}$ to both sides: $ \begin{eqnarray} x - 10 &=& \dfrac{5}{4} \\ \\ {+ 10} && {+ 10} \\ \\ x &=& \dfrac{5}{4} + 10 \end{eqnarray} $ Change the ${ + 10}$ to an equivalent fraction with a denominator of $4$ $ x = \dfrac{5}{4} {+ \dfrac{40}{4}} $ $ x = \dfrac{45}{4} $ Thus, the correct answer is $x = \dfrac{35}{4} $ or $x = \dfrac{45}{4} $.